Analytical Derivation of Steady-State Soil Water Probability Density Function Coupled with Simple Stochastic Point Rainfall Model
Publication: Journal of Hydrologic Engineering
Volume 13, Issue 11
Abstract
In this study, a new stochastic model for the propagation analysis of fluctuations in rainfall to soil water dynamics is proposed. Based on a lumped conceptualization of soil water dynamics with rainfall forcings, which are incorporated by a simple stochastic point rainfall model, a model is derived by using cumulant expansion theory from a stochastic differential equation. The advantage of the model is to provide the probabilistic solution in the form of a probability density function (PDF), from which one can find the ensemble average behavior of the system. Steady-state PDF of soil water is analytically obtained and analyzed for different climate, soil, and vegetation conditions. The city of Daegue in Korea, which represents the driest parts of the Korean, Peninsula, is applied for a case study and the result shows that the analytically derived steady-state soil water PDF can make a good agreement to the numerically obtained steady-state PDF from a lumped conceptualization model of soil water dynamics. From this agreement, the steady-state analysis is thought to be appropriate for the study of soil water dynamics where the seasonality of rainfall is not very significant. It is also shown that the fluctuations in rainfall tend to increase the variance of soil water dynamics, while the change of rainfall amount can shift the mode of PDF. General features for the PDFs as a function of different loss and soil characteristics are the decrease of soil water with loss rate and soil water storage capacity. The major conclusion, however, is that the proposed simplified stochastic soil water dynamic model for dry years in the Korean Peninsula can provide quite a reasonable explanation in the main soil water probabilistic properties when the rainfall variability is the only consideration.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This work was supported by Pukyong National University Research Fund (PKS-2005-006).
References
Budyko, M. I. (1974). Climate and life, Elsevier, New York.
Canadell, J., Jackson, R. B., Ehleringer, J. R., Mooney, H. A., Sala, O. E., and Schulze, E. D. (1996). “Maximum rooting depth of vegetation types at the global scale.” Oecologia, 108, 583–595.
Eagleson, P. (1978). “Climate, soil and vegetation: 1. Introduction to water balance dynamics.” Water Resour. Res., 14, 705–712.
Eagleson, P. (2002). Ecohydrology: Darwinian expression of vegetation form and function, Cambridge University Press, New York.
Entekhabi, D., and Rodriguez-Iturbe, I. (1994). “Analytical framework for the characterization of the space-time variability of soil moisture.” Adv. Water Resour., 17, 35–45.
Fernandez-Illescas, C. P., Porporato, A., Laio, F., and Rodriguez-Iturbe, I. (2001). “The ecohydrological role of soil texture in a water-limited ecosystem.” Water Resour. Res., 37, 2863–2872.
Fox, R. F. (1975). “A generalized theory of multiplicative stochastic processes using cumulant techniques.” J. Math. Phys., 16, 289–297.
Gardiner, C. W. (1985). Handbook of stochastic methods, Springer, New York.
Jackson, R. B., Canadell, J., Ehleringer, J. R., Mooney, H. A., Sala, O. E., and Schulze, E. D. (1996). “A global analysis of root distribution for terrestrial biomes.” Oecologia, 108, 389–441.
Kavvas, M. L. (2003). “Nonlinear hydrologic processes: Conservation equations for determining their means and probability distributions.” J. Hydrol. Eng., 8, 44–53.
Kavvas, M. L., and Karakas, A. (1996). “On the stochastic theory of solute transport by unsteady and steady groundwater flow in heterogeneous aquifers.” J. Hydrol., 179, 321–351.
Kim, S., Kavvas, M. L., and Chen, Z. (2005). “A root water uptake model under heterogeneous soil surface.” J. Hydrol. Eng., 10, 160–167.
Kubo, R. (1962). “Generalized cumulant expansion method.” J. Phys. Soc. Jpn., 17, 1100–1120.
Laio, F., Porporato, A., Fernandez-Illescas, C. P., and Rodriguez-Iturbe, I. (2001). “Plants in water-controlled ecosystems: Active role in hydrologic processes and response to water stress. II: Probabilistic soil moisture dynamics.” Adv. Water Resour., 24, 707–723.
Porporato, A., Daly, E., and Rodriguez-Iturbe, I. (2004). “Soil water balance and ecosystem response to climate change.” Am. Nat., 164, 625–633.
Rodriguez-Iturbe, I., Gupta, V. K., and Waymire, E. (1984). “Scale consideration in the modeling of temporal rainfall.” Water Resour. Res., 20, 1611–1619.
Rodriguez-Iturbe, I., and Porporato, A. (2004). Ecohydrology of water-controlled ecosystems: Soil moisture and plant dynamics, Cambridge University Press, New York.
Rodriguez-Iturbe, I., Porporato, A., Laio, F., and Ridolfi, L. (2001). “Plants in water-controlled ecosystems: Active role in hydrologic processes and response to water stress. I: Scope and general outline.” Adv. Water Resour., 24, 697–705.
Rodriguez-Iturbe, I., Porporato, A., Ridolfi, L., Isham, V., and Cox, D. R. (1999). “Probabilistic modeling of water balance at a point: The role of climate, soil and vegetation.” Proc. R. Soc. London, Ser. A, 455, 3789–3805.
Schulze, E. D., et al. (1996). “Rooting depth water availability and vegetation cover along an aridity gradient in Patagonia.” Oecologia, 108, 503–511.
Van Kampen, N. G. (1974). “A cumulant expansion for stochastic linear differential equations. II.” Physica (Utrecht), 74, 239–247.
Van Kampen, N. G. (1976). “Stochastic differential equations.” Phys. Rep., 24, 171–228.
Vanmarcke, E. (1983). Random fields: Analysis and synthesis, MIT Press, Cambridge, Mass.
Yoo, C., Kim, S., and Kim, T.-W. (2006). “Assessment of drought vulnerability based on the soil moisture PDF.” Stochastic Environ. Res. Risk Assess., 21, 131–141.
Yoo, C., Kim, S.-J., and Valdes, J. B. (2005). “Sensitivity of soil moisture field evolution on rainfall forcing.” Hydrolog. Process., 19, 1885–1869.
Yoon, J., and Kavvas, M. L. (2004). “Probabilistic solution to stochastic overland flow equation.” J. Hydrol. Eng., 8, 54–63.
Information & Authors
Information
Published In
Journal of Hydrologic Engineering
Volume 13 • Issue 11 • November 2008
Pages: 1069 - 1077
Copyright
© 2008 ASCE.
History
Received: Mar 12, 2007
Accepted: Jan 22, 2008
Published online: Nov 1, 2008
Published in print: Nov 2008
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.